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I am having difficulty drilling a Lucite stem (vucanite also). The bit tends to bind with the cut material. This binding generates heat very quick and kinda chews the hole internals. I am using a lathe at the lowest speed. Drill a little, back out, repeat. It binds about every < 3/16 inch going in. It binds when backing out no matter how small the advance increment.

I would try turning up the speed.

if the bit isn't cleaning itself for the first 1/4 inch or so you need a different bit

Lucite is very prone to melt suddenly and weld itself to the bit, resulting in either that chatter you noticed, breaking the lucite, or breaking the bit.

DO NOT increase drilling speed unless you really like melted lucite. Get a new sharp bit, and lube the hell out of it, clearing chips every 1/4 to 3/8 inch by pulling the bit completely. Some kind of cheap water-based lube, like soapy water for instance, really helps. I keep a spray bottle handy while drilling lucite, and spray both the bit and the hole. Windex is a good lube too, and if all else fails, an acrylic-safe cutting oil would probably be the best.

Since the tailstock of my lathe is warped and I don't have the skill to fix it, I drill freehand. Using a VSR drill, I think I drill at around 250-300rpm. I have to be careful not to let the bit ever stop in the lucite or it will melt to the plastic. Drill sharpness and lubrication seems to be the key. And that's sharp all the way up the flutes, not just on the cutting lips. If it's not shredding your fingertips or toothbrush when you clear the chips it's not sharp enough.

If either the tool gets hot or the lucite gets hot, you are basically screwed. Some kind of lube, or just drill a ways and then stop and let everything cool. The stuff works okay once you get a handle on it, but until you figure it out, it's awful to work with. Heat is your enemy.

In another thread, someone suggested keeping an ice-water soaked rag handy to quickly cool off the bit. It works great on ebonite, so it might help.

I don't work with acrylic much because I really hate how it behaves, but I do know that a lot of pen turners who use the stuff (as well as polyester resins) swear by Butter Flavor Pam as a drilling lubricant. Apparently regular Pam doesn't have the stuff needed.

You could always try some nose oil. I hear there's a good supplier nearby.